Optimal. Leaf size=70 \[ -\frac {\tanh ^{-1}\left (\frac {x^{q/2} \left (2 a+b x^{n-q}\right )}{2 \sqrt {a} \sqrt {b x^n+c x^{2 n-q}+a x^q}}\right )}{\sqrt {a} (n-q)} \]
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Rubi [A]
time = 0.05, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {1927, 212}
\begin {gather*} -\frac {\tanh ^{-1}\left (\frac {x^{q/2} \left (2 a+b x^{n-q}\right )}{2 \sqrt {a} \sqrt {a x^q+b x^n+c x^{2 n-q}}}\right )}{\sqrt {a} (n-q)} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 1927
Rubi steps
\begin {align*} \int \frac {x^{-1+\frac {q}{2}}}{\sqrt {b x^n+c x^{2 n-q}+a x^q}} \, dx &=-\frac {2 \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {x^{q/2} \left (2 a+b x^{n-q}\right )}{\sqrt {b x^n+c x^{2 n-q}+a x^q}}\right )}{n-q}\\ &=-\frac {\tanh ^{-1}\left (\frac {x^{q/2} \left (2 a+b x^{n-q}\right )}{2 \sqrt {a} \sqrt {b x^n+c x^{2 n-q}+a x^q}}\right )}{\sqrt {a} (n-q)}\\ \end {align*}
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Mathematica [F]
time = 0.44, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{-1+\frac {q}{2}}}{\sqrt {b x^n+c x^{2 n-q}+a x^q}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{-1+\frac {q}{2}}}{\sqrt {b \,x^{n}+c \,x^{2 n -q}+a \,x^{q}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^{\frac {q}{2}-1}}{\sqrt {b\,x^n+a\,x^q+c\,x^{2\,n-q}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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